## Journal of Symbolic Logic

### Ideals without CCC

#### Abstract

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F $\subseteq$ P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X $\rightarrow$ X with $f^{-1}[\{x\}] \not\in$ I for each x $\in$ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B $\not\in$ I and a perfect set P $\subseteq$ X for which the family $\{B + x : x \in P\}$ is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) $\Rightarrow$ (M) $\Rightarrow$ (B) $\Rightarrow$ not-ccc can hold. We build a $\sigma$-ideal on the Cantor group witnessing (M) & $\neg (D)$ (Section 2). A modified version of that $\sigma$-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for "nicely" defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 6).

#### Article information

Source
J. Symbolic Logic, Volume 63, Issue 1 (1998), 128-148.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745462

Mathematical Reviews number (MathSciNet)
MR1610790

Zentralblatt MATH identifier
0899.03035

JSTOR